We show a construction for a directed clique-width 3-expression for every \(G\in DDH\)  and then argue, why this is best possible.
The method we use is closely related to the undirected case: Distance-hereditary graphs have clique-width at most 3, see [1]}.
Let \(G\in \text{DDH}\)  and \(S = (s_1, \dots , s_n)\)  be a directed pruning sequence creating \(G\) .
We give an algorithm to construct a 3-expression traversing \(S\)  starting with the last element of the sequence.
The idea for computing this expression is to use three labels \(1,2\)  and 3 as follows: After every step of the algorithm, expressions which are already constructed consist of vertices labeled with 1 and 3, where 1 means that the vertex has not been finally treated and possibly has edges to other vertices not inserted yet, whereas for vertices labeled by 3 all incident edges have already been considered. The label 2 is only used as a working label.
For initialization, let \(X_v= \bullet _1\)  for every vertex \(v \in V\) . As this is a 1-expression, it is also a 3-expression.
In the following let now \(s_i = (v_i, op_i, a_i)\) , for simplification denoted by \((v, op, a)\) , be the currently treated element of \(S\)  and \(X_v\)  and \(X_a\)  be the 3-expressions which exists by induction for \(v\)  and \(a\) . Then, we get a 3-expression by the following rules depending on the operation \(op\) .
